Coupled oscillators
A weak spring between two simple gravity pendula causes a slight detuning in their proper frequencies, resulting in interference of two oscillatory motions of close but different speeds. In the difference frequency one can observe periodic energy interchange between the two pendula. A part of energy is also carried by the spring itself. In this demo you can try various mass ratios and the spring constants, as well as dragging the bobs directly.
Notice these effects:
- Apart from the difference frequency, also note what happens at the sum of the frequencies.
- Total energy is conserved in this system, but using any controllers one effectively gives the Hamiltonian an explicit time dependency and as such can pump or deplete energy from the system. Also hitting the top plane is intentionally inelastic.
- No linearization has been employed, so at higher deflections considerable distortion of the harmonics can be appreciated.
Tips for trying:
- Reducing the amplitudes to a level at which a small-angle approximation is appropriate. (The y-scale of the graph will adapt.)
- Making the two pendula swing in phase or antiphase.
- Stopping the pendula and then moving one of them very slowly (adiabatically) or abruptly (diabatically). Can you do so without causing the other one to oscillate?
- What conditions (mass ratio and spring constant) are the most favourable for perfect energy transfer from one pendulum to the other?